Psychophysical Neuroeconomics of Decision Making: Nonlinear Time Perception Commonly Explains Anomalies in Temporal and Probability Discounting ()
1. Introduction
Canonical representations on Hermitian symmetric spaces 
were introduced by Vershik-Gelfand-Graev [1] (for the Lobachevsky plane) and Berezin [2]. They are unitary with respect to some invariant non-local inner product (the Berezin form). Molchanov’s idea is that it is natural to consider canonical representations in a wider sense: to give up the condition of unitarity and let these representations act on sufficiently extensive spaces, in particular, on distributions. Moreover, the notion of canonical representation (in this wide sense) can be extended to other classes of semisimple symmetric spaces
, in particular, to para-Hermitian symmetric spaces, see [3]. Moreover, sometimes it is natural to consider several spaces 
together, possibly with different
, embedded as open 
-orbits into a compact manifold
, where 
acts, so that 
is the closure of these orbits.
Canonical representations can be constructed as follows. Let 
be a group containing 
(an overgroup), 
a series of representations of 
induced by characters of some parabolic subgroup 
associated with 
and acting on functions on
. The canonical representations 
of 
are restrictions of 
to
.
In this talk we carry out this program for para-Hermitian symmetric spaces of rank one. These spaces are exhausted up to the covering by spaces 
with
,
. For these spaces
, an overgroup is the direct product 
and canonical representations turn out to be tensor products of representations of maximal degenerate series and contragredient representations. These tensor products are studied in [4], see also [5]. So we lean essentially on these papers [4,5]. We decompose canonical representations into irreducible constituents and decompose boundary representations. Notice that in our case the inverse of the Berezin transform 
can be easily written: precisely it is the Berezin transform
.
Canonical and boundary representations for 
in the case 
(then 
is the hyperboloid of one sheet in
) were studied in [6]. For the two-sheeted hyperboloid in
, it was done in [7].
In this paper we present only the main results. The detailed theory of canonical and boundary representations, for example, on a sphere with an action of the generalized Lorentz group, can be seen in [8].
Let us introduce some notation and agreements.
By 
we denote
. The sign 
denotes the congruence modulo 2.
For a character of the group 
we shall use the following notation
where
, 
,
.
For a manifold
, let 
denote the Schwartz space of compactly supported infinitely differentiable 
-valued functions on
, with a usual topology, and 
the space of distributions on
—of anti-linear continuous functionals on
.
2. The Space 
and the Manifold 
We consider the symmetric space 
where
, 
,
.
The group 
acts on the space 
by
Let us write matrices in 
in block form according to the partition 
of
. Let us take the matrix
The subgroup 
is just the stabilizer of this point
, this subgroup consists of block diagonal matrices:
Thus, our space 
is the 
-orbit of
, it consists of matrices of rank one and trace one.
Equip 
with the standard inner product
, let
. Let 
be the sphere
. Let 
be the Euclidean measure on
. The group 
acts on 
by
.
Let 
be a cone in 
consisting of matrices 
of rank one. Therefore, the space 
is the section of 
by the hyperplane
.
Introduce a norm 
in 
by
where the prime denotes matrix transposition.
Let 
be the section of 
by
.
Define a map 
by
It is a two-fold covering. The measure 
defines a measure 
on 
by
The action of the group 
on 
gives the following action of 
on
:
In particular, the subgroup
, a maximal compact subgroup, acts on 
by translations:
Let us consider on 
the function
(1)
The action on 
has three orbits: namely, two open orbits (of dimension
): 
and 
and one orbit of dimension
: 
. The orbit 
is a Stiefel manifold, it is the boundary of
. Denote
. Each of orbits 
can be identified with the space
. The map is constructed by means of generating lines of the cone
.
3. Maximal Degenerate Series Representations
Recall [4] maximal degenerate series representations
, 
, 
, of the group
. Let 
be the subspace of 
consisting of functions 
of parity
:
. The representations 
act on 
by
4. Representations of 
Associated with 
Recall [5] a series of representations 
of the group 
associated with the space
.
Denote by 
the space of functions 
in 
of parity
:
The representation 
acts on 
by
(2)
Let 
denote the following sesqui-linear form
(3)
Define an operator 
on 
by
It intertwines 
and
. The operator 
is a meromorphic function of
. Let us normalize this operator (multiplying it by a function of
) such that the normalized operator 
is an entire non-vanishing function of
.
There are three series of unitarizable irreducible representations. The continuous series consists of 
with
, 
, the inner product is (3). The complementary series consists of 
with
, the inner product is 
with a factor. The discrete series consists of the representations 
where
,
, 
, which are factor representations of
on the quotient spaces
. The representations 
with the same 
and different
are equivalent. It is convenient to take 
where 
for odd 
and 
for even
. The inner product is induced by the form
.
5. Canonical Representations
We define canonical representations
, 
, 
, of the group 
as tensor products:
They can be realized on
: let 
denote the subspace of 
consisting of functions 
of parity
:
, then the representation 
acts on 
by a formula similar to (2):
The inner product
(4)
is invariant with respect to the pair
, i.e.
(5)
Consider an operator 
on 
defined by
It turns out that the composition 
is equal to the identity operator 
up to a factor. We can take 
such that
namely,
With the form (4) the operator 
interacts as follows:
(6)
This operator 
intertwines the representations 
and
, i.e.
Let us call it the Berezin transform.
Let 
be the space of distributions on 
of parity
. We extend 
and 
to 
by (5) and (6) respectively and retain their names and the notation.
Let us introduce the following Hermitian form 
on
:
Let us call this form the Berezin form.
6. Boundary Representations
The canonical representation 
gives rise to two representations 
and 
associated with the boundary 
of the manifolds 
(boundary representations). The first one acts on distributions concentrated at
, the second one acts on jets orthogonal to
.
We can introduce “polar coordinates” on 
corresponding to the foliation of 
into 
-orbits. The 
- orbits are level surfaces of the function
, see (1). For 
the 
-orbits are diffeomorphic to
. In these coordinates the measure 
on 
is
where 
is the measure on
.
Let 
be a function in
. Consider it as a function of polar coordinates. Consider its Taylor series 
in powers of
. Here 
are functions in
. Denote by
, 
, 
, the space of distributions in
, having the form
where
, 
is the Dirac delta function on the real line, 
its derivatives. Let 
.
Denote by 
Taylor coefficients of the function
. The distribution 
acts on a function 
as follows:
(7)
Denote by 
the restriction of 
to
. This representation is written as a upper triangular matrix with the diagonal
,
.
Distributions in 
can be extended in a natural way to a space wider than
. Namely, let 
be the space of functions 
of class 
on 
and 
of parity 
and having the Taylor decomposition of order
:
where
. Then (7) keeps for 
with
.
Let 
denote the column of Taylor coefficients
. The representation 
acts on these columns:
It is written as a lower triangular matrix with the diagonal
,
.
The boundary representations 
and 
are in a duality.
7. Poisson and Fourier Transforms
Let us write operators 
and 
intertwining representations 
and
. We call them Poisson and Fourier transforms associated with canonical representations.
The Poisson transform 
is a map 
given by
It intertwines 
with
. Here we consider
as the restriction to 
of the representation 
acting on distributions in
.
For a 
-finite function 
and 
the Poisson transform has the following decomposition in powers of
:
where 
has polar coordinates
. Here 
and
are certain operators acting on
. The factors 
and 
give poles of the Poisson transform in 
depending on
:
(8)
where 
and
,
. If a pole belongs only to one of series (8), then the pole is simple, and if a pole belongs to both series (8), then 
and the pole is of the second or first order.
Let the pole
, 
, be simple. The residue 
of 
at this pole is an operator
. Denote the image of this operator by
.
The Fourier transform 
is a map 
given by
It intertwines 
with
.
The Fourier and Poisson transforms are conjugate to each other:
Poles in 
of the Fourier transform are situated at points
(9)
where 
and
,
. If a pole belongs only to one of the series (9), then the pole is simple, and if a pole belongs to both series (9), then 
and the pole is of the second or first order.
Let the pole
, 
, be simple.
The residue 
of 
at this pole is a “boundary” operator
,
. The operator 
is defined in terms of Taylor coefficients
: it is a linear combination of functions
. Therefore, we may consider the following operator 
acting on columns
of functions
: this operator to any column 
assigns the column
of functions in the same space
—by the same formulas without
. This operator 
is given by a lower triangular matrix.
8. Decomposition of Boundary Representations
The meromorphic structure of the Poisson and Fourier transforms is a basis for decompositions of boundary representations 
and
.
Let the pole 
of the Poisson transform is simple, in particular, it happens when
. Then the boundary representation 
is diagonalizable which means that 
decomposes into the direct sum of
, and the restriction of 
to 
is equivalent to 
(by means of
).
If a pole is of the second order, then the decomposition of 
contains a finite number of Jordan blocks, this number depends on
.
Let the pole 
of the Fourier transform is simple, in particular, when
. Then the matrix 
is diagonalizable which means that
is a diagonal matrix. Its diagonal is
,
.
If a pole is of the second order, then the decomposition of 
contains a finite number of Jordan blocks, this number depends on
.
9. Decomposition of Canonical Representations
Let us write decomposition of canonical representations. We restrict ourselves to a generic case: 
lies in strips
Case (A):
.
Theorem 1 Let
. Then the canonical representation 
decomposes—as the quasiregular representation [5]—into irreducible unitary representations of continuous and discrete series with multiplicity one. Namely, let us assign to a function 
the family of its Fourier components
, 
, 
, 
, and
,
. This correspondence is 
equivariant. There is an inversion formula:
(10)
and a “Plancherel formula” for the Berezin form:
(11)
Here 
and 
stand for the Plancherel measure for
, see [5], the factor 
is given by following formula:
Case (B):
.
Here we continue decomposition (10) analytically in 
from 
to
,
. Some poles in 
of the integrand intersect the integrating line—the line 
. They are poles 
and 
of the Poisson transform with
. They give additional summands to the right hand side. So after the continuation we obtain:
(12)
where the integral and the series mean the same as in (10) and
are some numbers.
Similarly, the continuation of (11) gives
(13)
where the integral and the series mean the same as in (11) and 
are some numbers.
The operators
, 
, can be extended from
to the space 
and therefore to the sum
Then these operators 
turn out to be projection operators onto
. Moreover, there are some “orthogonality relations” for them. Decomposition (13) can also be extended to the space
. This decomposition is a “Pythagorean theorem” for decomposition (12).
Theorem 2 Let
,
. Then the space 
has to be completed to
. On this space the representation 
splits into the sum of two terms: the first one decomposes as 
does in Case (A), the second one decomposes into the sum of 
irreducible representations
,
. Namely, let us assign to any 
the family
where
, 
,
. This correspondence is 
-equivariant. There is an inverse formula, see (12), and a “Plancherel formula”, see (13).
Case (C):
.
Now we continue decomposition (10) analytically in 
from 
to
. Here poles 
and
, 
, 
, of the integrand (they are poles of the Fourier transform) give additional terms. We obtain
(14)
where the integral and the series mean the same as in (10) and
some numbers. The operators 
can be extended to the space
,
. Denote by 
the image of
. It turns out that the operators 
are projection operators onto 
and for them there are some “orthogonality relations”.
Now we continue decomposition (11) from 
to
. Poles of the integrand which intersect the integrating line 
and give additional terms (they are poles of both Fourier transforms) turn out fortunately to be of the first order, since at these points the function 
as a function of 
has zero of the first order. After the continuation we obtain:
(15)
where the integral and the series mean the same as in (11), 
some numbers. It is a “Pythagorean theorem” for decomposition (14).
Theorem 3 Let
,
. Then the representation 
considered on the space 
splits into the sum of two terms. The first one acts on the subspace of functions 
such that their Taylor coefficients 
are equal to 0 for 
and decomposes as 
does in Case (A), the second one decomposes into the direct sum of 
irreducible representations
, 
acting on the sum of the spaces
. There is an inversion formula, see (14), and a “Plancherel formula” for the Berezin form, see (15).